We study the effect of inhomogeneities on light propagation. The Sachs
equations are solved numerically in the Swiss-Cheese models with
inhomogeneities modelled by the Lemaitre-Tolman solutions. Our results imply
that, within the models we study, inhomogeneities may partially mimic the
accelerated expansion of the Universe, but the effect is small.

There have been several papers on the effect of inhomogeneities on light propagation using Swiss Cheese models recently. The idea is to cut spherical holes from a FRW universe and replace the contents with another spherical solution. Historically, this was the Schwarzschild solution, so the holes were empty except for a central mass. From the junction conditions, it then follows that the hole expands on average in the same way as the FRW solution, so average evolution of the spacetime does not change, only the optical properties. However, if the interior solution is a general spherically symmetric dust solution, the Lemaitre-Tolman-Bondi (LTB) model, the average evolution can be different from that of the FRW background.

This paper is the most careful study of Swiss Cheese with LTB holes thus far. All of the junction conditions are taken into account, the holes have realistic sizes, the entry angle is randomised, the centers of the holes are not assumed to lie on the same plane and the holes do not overlap. The study of the null shear (usually neglected) is a nice touch too – the results show that neglecting the null shear when calculating the angular diameter distance is justified.

For holes of realistic size, the quoted change in the distance out to redshift z = 1.5 in a background Einstein-de Sitter universe is about 10% of introducing a cosmological constant with ΩΛ0 = 0.7. The spherical nature of the holes, density profile chosen by hand and so on means that the model is not quite realistic, but even so it shows that one can have a large effect. An interesting question is whether the change in light propagation can be understood here in terms of changes in the average quantities, as argued in 0812.2872 and 0912.3370. (The paper seems to imply that the average quantities would be the same as in the FRW case, but this may not be the case.)

There is a new version of the paper. The entry angles are now randomised correctly for balls randomly positioned in three-dimensional space. The distribution is flat neither in the entry angle nor the impact parameter - I think this is the first paper to have the correct distribution for the entry angle. (At least recently - I'm not sure about sure about some older papers from the 90s and earlier.)

The previous result of a 10% effect (compared to ΛCDM) turns out to be the result of correlations betwee the holes implicit in the earlier assumption of a flat distribution of the impact parameter. Done correctly, the effect is negligible, which is expected, given that the mean density along the light ray is close to the spatial average of the density, which in turn is close to the FRW value.